&RDUVH *UDLQHG Molecular Dynamics
A tool in biophysical chemistry
Material Properties
Statistical Thermodynamics
Potentials & Distribution
Meaning of Time
Numerical Approximations
Material Properties
› Matter is characterized by
• Structure
• Spatial relationships between building blocks
• ORGANIZATION
• Dynamics
• Temporal behavior of structure
• FLUCTUATIONS
• RESPONSE
(Molecular) Driving Forces
› Material properties are due to interactions between
building blocks
• ENERGY
• Electrostatic, dispersion, repulsion
• Chemical bond
› and heat
• Fundamental restlessness of nature
• ENTROPY
• Counts the accessible states of the system
• Temperature
Ingredients for Molecular Modeling
Van Gunsteren et al. Angew. Chem. Int. Ed. 45, 4064 (2006)
Statistical Thermodynamics
› Enables computation of (macroscopic) properties
• in terms of INTERACTIONS
• between (microscopic) building blocks for systems containing
many building blocks
• Quantified in energy function with variables
• positions (R) and momenta (P)
• GIVEN CERTAIN BOUNDARY CONDITIONS
• controlling the system
• Constant number of particles OR chemical potential
• Constant volume OR pressure
• Constant energy OR temperature
Phase Space
› Describes possible states of the system
• Dimension 2*Ndof x 2*Ndof (R,P) or Ndof x Ndof (R)
• A single system, in time, follows an allowed path
through phase space visiting allowed states (R,P)
Q = ! wi
Partition Function
› Describes appropriate probabilities of the
possibilities of the system being realized
• determined by boundary conditions
• KNOWLEDGE OF PROBABILITIES
• Enables calculation of properties
A= A
wA
!
=
!w
i
i
i
1
= ! wi Ai
Q
Partition Function of Macroscopic Systems
› Depends on potential (internal) energy V
• By building ensembles of coupled replicas of the
system of interest it can be shown that
w ( R ) = f (V ( R ))
• Potential is of central interest!
• For constant Temperature
w (R) =
(
exp !V ( R ) kBT
Q
);
(
Q = " dR exp !V ( R ) kBT
)
Potential and Distribution
› Close link between potential and distribution
V (R)
w (R)
R
• Can be used to extract potential from structural data
w (R) =
(
exp !V ( R ) kBT
Q
) " V ( R ) = !k T ln #w ( R )Q %
B
$
• Inverse Boltzmann techniques
• In MARTINI for fine-tuning bonded interactions
&
Potential and Time
› Close link between potential and frequency
(
V (R) = 1 2 K R ! R0
)
M
2
Etot
R
• Time required to traverse potential depends on shape
of the potential and the mass
• For harmonic oscillator, the characteristic frequency
! = (K M )
1/2
• Can be used to gauge time
Equations of Motion
› Distribution and characteristic frequency obtained by
following the system in time
M
!! = F = ! "V ( R ) "R
MR
R
Etot
! ( 0 ) !t + F ( 0 ) ( !t )2 2M
R ( !t ) " R ( 0 ) + R
• Propagate in time from a starting point according to
Newtonian equations of motion
• Conservation of energy is important because
otherwise we get wrong distribution and frequency
! + F ( !t )2 2M
!R = R!t
Basic MD Algorithm
› Numerically integrate Newton’s Law: F = m*a
› t=0
› Forces &
velocities
› displacements
› t=1
› new displacements, etc...
Conservation of Energy
› In numerical approaches, time-step should be such that
PES* is followed
!! = F = ! "V ( R ) "R
MR
F ( !t ) 2M
2
› We should end up at the
energy we expect or energy
is lost from/added to the
system
›
*Potential Energy Surface: multidimensional surface that gives the energy as
a function of all co-ordinates
Conservation of Energy
› In practice, even with small time-steps, conservation of
energy is difficult to achieve
!! = F = ! "V ( R ) "R
MR
! ! = (1 + " )R
!
R
F ( !t ) 2M
2
› Couple to heat bath to dissipate or gain heat
› Popular methods are velocity rescaling (e.g. Berendsen) or
extended ensembles (e.g. Nose-Hoover)
› Strictly speaking, we are not doing Newtonian mechanics
› Time step and bath couling strengths are part of the
parameter set!
Cut-off noise
› Particles start/stop interacting at certain
distance: energy is not conserved
Rc
Rc
Sampling
› Accuracy of distribution depends on sampling
M
M
R
Etot
w (R)
• Smaller time step leads to better sampling
• Rule of thumb: 5-10 points on a harmonic potential characterize
it reasonably well
Appropriate time step for sampling
› At atomistic level, time step for fastests vibrations
M
1 / !t vib = " IR = cv!IR #
R
3 *10 8 * 3, 000 %& ms $1cm $1 '(
# 1*1014 Hz
• Time step ~1 fs
w (R)
Etot
Aim and Scope of Simulation
› For larger systems, PES is very complicated
• Aim is to obtain a REPRESENTATIVE SAMPLE of the
REALIZATIONS of the system
• Structural and/or dynamic
› Phase space (R, P) to be sampled depends on
• Degrees of freedom
• WE DEFINE THE SYSTEM OF INTEREST AND
CHOOSE A MODEL TO YIELD INFORMATION ON
A CERTAIN LENGTH AND TIME SCALE
• Realism depends on judiciousness of our choice
Molecular Models Considered
› (United) Atoms
• Each atom (or CHn group) is treated as a particle
› Coarse grained superatoms (MARTINI)
• A group of (~4 united atoms) is treated as a particle
› DPPC molecule (a lipid) in UA and
CG (MARTINI) representations
COARSE GRAINING BENEFITS
› Reduced Complexity
• Physics: not all detail is relevant for our question
› Efficiency: increase length and time scales
• Space: reduced density reduces number of interactions
• 4 (number) x 4 (pairs)
• 4 neighbor searching
• Time: smoother energy landscape, increased time step
• in algorithm: 10-20
• effective time: 4
• Total MARTINI: 2.5-5 103 speed-up compared to UA
SUMMARY
› Molecular Dynamics
• Generate representative ensemble of structures
connected through representative dynamics
› Molecular Model
• Should reflect research question
• Potentials determine distributions and dynamics
• Complexity increases with level of detail
• Rougher potential energy surface
• Larger phase space
• Slower sampling
Appropriate Interaction Potentials
› Study structure through distribution functions
• Find that in liquid state we need fairly short-ranged potentials
• Best correspondence by numerical potentials obtained via
Boltzmann Inversion techniques
• For convenience, simple analytical forms are often used
(variations of the Lennard-Jones potential)
Pictures: A. Lyubartsev Eur Biophys. J. 35, 53-61 (2005)
V ( R ) = !k BT ln "# w ( R ) Q $%
Appropriate Interaction Potentials
› Study structure through distribution functions
• Find that in liquid state we need fairly short-ranged potentials
Boltzmann
inversion
• No clear structure beyond ~1.0 nm
• Similar findings for many liquid-state systems from experiment
Pictures: A. Lyubartsev Eur Biophys. J. 35, 53-61 (2005)
Appropriate Interaction Potentials
› Approximate potentials
• Compare IBP to common analytical potentials
• Approximations may not be too bad
Coarse-graining Philosophies
› HIERARCHICAL MODELING
• Part of a wider scheme
• Interactions at less detailed level are the result of
the collective interactions at more detailed level
• General method applicable to any system (like an
algorithm)
› EFFICIENT MODEL AT CERTAIN SCALE
• Reproduce faithfully certain chosen properties
• Developed with certain application in mind
• Nevertheless aiming at wide use through
considering the physics of the problems in mind
H.J.C. Berendsen Simulating the Physical World Cambridge University Press Hierarchical modeling: pros and cons
› PRO
• UNBIASED
• Physics follows through the hierarchy of models
• STRAIGHTFORWARD MULTISCALING
• Enables reliable combination of levels of modeling
• Entirely general approach
› CON
• REQUIRES LARGE WORK LOAD
• Need detailed level simulations to derive CG potentials
• Complicated numerical potentials
• LIMITED VALIDITY
• Strictly valid for one state point only (new system, new
potentials)
Multiscaling
› Compatible models at different length and time
scales
• Combine them in one system
• Space resolution (e.g. QM-MM)
• Resolution exchange (cf. Replica exchange)
• Switch between resolutions
• Propagate at coarse level to generate an ensemble
• Zoom in on snapshots and add detail
› What is compatible?
• Phase space at less detailed level matches phase space at
detailed level mapped to the less detailed level
G. Voth, Ed Coarse-graining of Condensed Phase and Biomolecular Systems Semi-empirical modeling: pros and cons
› PRO
• CHEAP
• Parameterize on empirical data available
• Simple analytical potentials
• TRANSFERABLE
• After parameterizing building blocks, many similar systems
can be treated straightforwardly
› CON
• BIASED
• Toward parameterized properties
• PROBLEMATIC MULTISCALING
• Different levels do not need to correspond closely
• Extensive validation required
Semi-empirical modeling: partitioning
› MARTINI and GROMOS 53A6 Philosophy
• Partitioning properties are important in biomolecular
modeling
Semi-empirical modeling: parameterization
› SIZE
• From density
• Partial volumes
› COHESION
• From heat of vaporization
› PARTITIONING
• From free energy of solvation
• In water
• In alkane
• Everything in everything
Partitioning from Simulations
› Indirect by restricted sampling
› Direct by counting
+
+
!G12 =
+
+
#
2
1
d"
( )
$H "
$"
K12 =
› Thermodynamic integration
c2
c1
=
N 2 V1
N1 V2
!
p2
p1
;!G12 = RT ln K12
MARTINI modeling: potentials
› NON-BONDED
• Modified Lennard-Jones and Coulomb
Smooth transition at
cut-off to reduce cut-off
noise can be achieved in
different ways.
Potential and Force
vanish at cut-off!
S.J. Marrink et al. J. Phys. Chem. B. 108, 750-760 (2004)
R. Baron et al. Phys. Chem. Chem. Phys. 8, 452-461 (2007)
Appropriate Interaction Potentials
› Bonds and angles
• At UA level, bonds and angles aptly described by
harmonic potentials (narrow Gaussian distributions,
e.g. from X-ray scattering experiments)
• MARTINI model also uses these, but with much lower
force constants
MARTINI Potentials
› BONDED: Simple harmonic/cosine bonds & angles
› Torsional potentials are not used in standard MARTINI
• Can be added, but long harmonic bonds more stable
COARSE GRAINING CAVEATS
› A SIMPLE EXAMPLE
• 2-to-1 mapping scheme of harmonic oscillator
• Weak coupling through collisions (gas)
› Weak LJ
potential
› Harmonic
springs
› Small mass
› Large mass
COARSE GRAINING CAVEATS
› A SIMPLE EXAMPLE
• 2-to-1 mapping scheme of harmonic oscillator
• Coarse grain on Center of Mass
COARSE GRAINING CAVEATS
› A SIMPLE
EXAMPLE
• A look at 100 ps
trajectory for the
system
• Energy exchange
through
collisions
COARSE GRAINING CAVEATS
› A SIMPLE EXAMPLE
• A look at 10 ps trajectory for one oscillator
COARSE GRAINING CAVEATS
› DETAIL IS LOST
• Distribution reflects effective interaction
• Can possibly be achieved by a simple potential
(
V (R) = K R ! R0
DERIVING POTENTIAL
› Harmonic force constant from Gaussian
distribution
(
w ( R ) ! exp # " K R " R 0
$
1
(
( dx ! " exp $% # x # x0
)
2
)
2
2kBT %
&
!2& =1
'
› Fit for best x0 and σ:
K = 2k BT !
2
)
2
2
COARSE GRAINING CAVEATS
› Compare distributions at CG level
• In this simple(?) example, there are some
complications
• Frequent
collisions
required to get
good statistics
COARSE GRAINING CAVEATS
› Smoother interaction, smoother motion
COARSE GRAINING CAVEATS
› THE MEANING OF TIME
• Smoother interaction, smoother motion
• Enables larger time steps
• Friction is lower, sampling speeds up
• Barriers are more easily overcome
› DETAIL IS LOST
• Physics may be disturbed
• Exchange of energy between modes less efficient
COARSE GRAINING CAVEATS
› THE MEANING OF TIME
• Re-introduce friction through stochastic term
• Part of benefit is gone, but dynamics are more
realistic
COARSE GRAINING CAVEATS
› THE MEANING OF TIME
• Effective time scale can be investigated by comparing
to more detailed-level simulations
• Diffusion
• Build-up of configurational entropy
• MARTINI model for lipids: effective time is
approximately 4 times simulation time
COARSE GRAINING SUMMARY
› SPEEDING UP SEARCH IN PHASE SPACE
• Use an effective interaction from distribution
• Smooths Potential Energy Surface
• Reduces number of interactions
• Increases effective time
› DETAIL IS LOST
• Beneficial: detail may not be required
• Detrimental: some system characteristics are gone